mathematical analysis of diffusion and kinetics of...

Applied and Computational Mathematics 2017; 6(3): 143-160

http://www.sciencepublishinggroup.com/j/acm

doi: 10.11648/j.acm.20170603.13

ISSN: 2328-5605 (Print); ISSN: 2328-5613 (Online)

Mathematical Analysis of Diffusion and Kinetics of Immobilized Enzyme Systems that Follow the Michaelis – Menten Mechanism

Krishnan Lakshmi Narayanan1, Velmurugan Meena

2, *, Lakshman Rajendran

1, Jianqiang Gao

3,

Subbiah Parathasarathy Subbiah4

1Department of Mathematics, Sethu Inistitute of Technology, Kariapatti, India 2Department of Mathematics, Madurai Kamaraj University Constitutional College, Madurai, India 3Department of Computer Information, Hohai University, Nanjing, China 4Mannar Thirumalai Naiker College, Pasumalai, Madurai, India

Email address:

[email protected] (K. L. Narayanan), [email protected] (V. Meena), [email protected] (L. Rajendran),

[email protected] (Jianqiang Gao), [email protected] (S. P. Subbiah) *Corresponding author

To cite this article: Krishnan Lakshmi Narayanan, Velmurugan Meena, Lakshman Rajendran, Jianqiang Gao, Subbiah Parathasarathy Subbiah. Mathematical

Analysis of Diffusion and Kinetics of Immobilized Enzyme Systems that Follow the Michaelis – Menten Mechanism. Applied and

Computational Mathematics. Vol. 6, No. 3, 2017, pp. 143-160. doi: 10.11648/j.acm.20170603.13

Received: April 5, 2017; Accepted: April 18, 2017; Published: June 21, 2017

Abstract: In this paper, mathematical models of immobilized enzyme system that follow the Michaelis-Menten mechanism

for both reversible and irreversible reactions are discussed. This model is based on the diffusion equations containing the non-

linear term related to Michaelis-Menten kinetics. An approximate analytical technique employing the modified Adomian

decomposition method is used to solve the non-linear reaction diffusion equation in immobilized enzyme system. The

concentration profile of the substrate is derived in terms of all parameters. A simple expression of the substrate concentration is

obtained as a function of the Thiele modulus and the Michaelis constant. The numerical solutions are compared with our

analytical solutions for slab, cylinder and spherical pellet shapes. Satisfactory agreement for all values of the Thiele modulus

and the Michaelis constant is noted. Graphical results and tabulated data are presented and discussed quantitatively to illustrate

the solution.

Keywords: Mathematical Modeling, Nonlinear Differential Equations, Modified Adomian Decomposition Method,

Michaelis-Menten Kinetics, Immobilized Enzyme

1. Introduction

Many problems in theoretical and experimental biology

involve reaction diffusion equations with nonlinear chemical

kinetics. Such problems arise in the formulation of substrate

and product material balances for enzymes immobilized

within particles [1] in the description of substrate transport

into microbial cells [2], in membrane transport, in the

transfer of oxygen to respiring tissue and in the analysis of

some artificial kidney systems [3]. For such cases, the

problem is often well poised as a two-point nonlinear

boundary-value problem because of the saturation,

Michaelis-Menten, or Monod expressions which are used to

describe the consumption of the substrate.

Mireshghi et al [4] provide a new approach for estimation

of mass transfer parameters in immobilized enzymes

systems. Benaiges et. al [5] studied the isomerzation of

glucose into fructose using a commercial immobilized

glucose-isomerase. The Michaelis-Menten equation is the

most common rate expression used for enzyme reactions.

This equation can also be used for immobilized enzymes [6,

7]. Many authors discussed the application of immobilized

enzyme reactors extensively, but immobilized enzyme

engineering is still in its infancy. Several general categories

144 Krishnan Lakshmi Narayanan et al.: Mathematical Analysis of Diffusion and Kinetics of Immobilized Enzyme Systems that

Follow the Michaelis – Menten Mechanism

of immobilized enzyme reactors such on: batch reactors,

continuous stirred reactors, fixed bed reactors and fluidized

bed reactors exist. When the immobilized enzyme is in the

form of spheres, chips, discs, sheets or pellets it can be packed

readily into a column [8-10].

Many authors presented enzymatic kinetics of irreversible

[11-15] and reversible [14-16] mechanism. Farhad et al. [15]

solved the nonlinear differential equations in enzyme kinetic

mechanism using finite difference method. To the best of our

knowledge, till date, no rigorous analytical expressions for the

steady-state concentration for immobilized mechanisms are

derived. We have presented the analytical expressions for

substrate concentrations for all the three cases of kinetic models

and for all possible values of the parameters using modified

Adomian decomposition method [16-21]. These results are

compared with the numerical results and are found to be good in

the agreement. A simple analytical expression of the

concentration is obtained for various particle shapes and for

external mass transfer resistance boundary condition. Also, the

general expressions for the mean integrated effectiveness factor

for all values of parameters are presented.

2. Formulation of the Problem and

Analysis

The immobilized enzyme systems may be considered to be

porous slab, cylinder and sphere shapes where enzymes are

uniformly distributed on the surface and in the interior.

Pictorial representation of the enzyme in the biocatalyst is

provided in Figure. 1.

Figure 1. Pictorial representation of the enzyme in the biocatalyst.

It is further assumed that: (1) The kinetics of the free

enzyme are described by the Michaelis-Menten equation for

irreversible reactions and by the modified Michaelis-Menten

equation for reversible reactions; (2) No partition effect

exists between the particle surface and the interior; (3) The

temperature, density, and effective diffusivity of reactants

inside the particle are constant; (4) A quasi-steady-state

condition is attained; (5). The partition effect between the

support and bulk fluid phase is neglected; (6) Enzyme

deactivation is neglected. Based on these above assumptions,

the governing differential equations with boundary

conditions for irreversible and reversible reactions are

reported below.

Nomenclature

Symbols Définitions Units

a Volume of the fluid phase in the

reactor none

Bi Biot number reaction in the pellet none

C

Dimensionless substrate

concentration for the reversible

reaction in the pellet

none

C

Dimensionless substrate

concentration for the reversible

reaction in the pellet

none

eD Effective diffusivity of the

substrate in the pellet 2 / mincm

Ef Effective factor none

g Pellet shape factor none

1k External mass-transfer co-efficient /cm s

mK Irreversible reaction Michaelis

constant 3/kg m

'mK

Reversible reaction Michaelis

constant 3/kg m

mfK Reversible reaction Michaelis

constant M

mrK Michaelis constant of the reverse

reaction M

0v Initial reaction rates none

mV Irreversible maximum reaction

rate 3/ /kg s m cat

mfV Reversible maximum reaction rate 3/ /kg s m cat

mrV Maximum velocity of the reverse

reaction / min/mol l cat

R Half-thickness of the pellet m

S Irreversible substrate

concentration inside the pellet 3/mol cmµ

eqS Equilibrium substrate

concentration 3/mol cmµ

S Reversible substrate concentration

inside the pellet 3/mol cmµ

bS

Irreversible substrate

concentration in the bulk fluid

phase

3/mol cmµ

0bS

Irreversible initial substrate

concentration in the bulk fluid

phase

3/mol cmµ

bS Reversible substrate concentration

in the bulk fluid phase 3/mol cmµ

0bS Reversible substrate concentration

in the bulk fluid phase 3/mol cmµ

t Time min

X Dimensionless distance none

x Distance to the center none

Y Dimensionless reversible substrate

concentration none

Greek

symbols

α Dimensionless parameter in

reversible reaction none

Applied and Computational Mathematics 2017; 6(3): 143-160 145

bβ

Dimensionless parameter in

irreversible reaction for bulk fluid

phase

none

bβ

Dimensionless parameter in

reversible reaction for bulk fluid

phase

none

0bβ Dimensionless parameter for

initial fluid phase none

φ Intrinsic modified Thiele modulus

in irreversible reaction none

φ Dimensionless parameter in

irreversible reaction none

2.1. Irreversible Reactions

A differential mass balance equation for the substrate for

irreversible reactions in dimensionless form can be

represented as follows [15]:

22

2

1

1 b

d C g dC C

X dX CdXφ

β−+ =

+ (1)

The boundary conditions are given by

0, 0dC

XdX

= = (2)

X 1, C 1= = (without external mass transfer resistance) (3)

1, (1 )dC

X Bi CdX

= = − (with external mass transfer resistance) (4)

where C represents the dimensionless substrate

concentration, X represents the dimensionless distance to

the center or the surface of symmetry of the pellet, φ , bβ

and Bi represents the Thiele module, dimensionless

parameter for bulk fluid phase and Biot number

respectively. The g characterizes the shape of the

immobilized catalyst with g =1, 2, 3 for a slab, cylindrical,

and spherical pellets respectively. So it can be regarded as a

‘shape factor’ for the particle. The dimensionless variables

are defined as follows:

, , ,b mb

b m m e

S VS xC X R

S R K K Dβ φ= = = = (Theile modulu

s), 1

e

KBi R

D= (Biot number) (5)

In the above expressions, the parameters

1,, , , ande bx R K D S S represent the distance to the center,

the half-thickness of the pellet, the external mass transfer

coefficient, the effective diffusivity of the substrate in the

pellet, the substrate concentration inside the pellet and

substrate concentration in the bulk fluid phase respectively.

andm mK V are the kinetic parameters. The equation (1) also

describes the temperature or concentration variation in many

fields of physics, chemistry, biology, biochemistry, and many

others [7–14]. The effectiveness factor, Ef is given by

( ) ( )1

1

0

11

gb

b

CEf g X dX

Cβ

β−= +

+∫ (6)

The initial substrate reaction rate 0v is given by

00

0

m b

m b

V Sv Ef

K S=

+ (7)

where 0bS denotes the initial substrate concentration.

2.2. Reversible Reactions

For reversible reactions, the governing differential

equation for the dimensionless substrate concentration in the

pellets is the same as Eq. 1 except the following

dimensionless parameters.

, , ,b mb b eq

m m eb

S VSC R S S S

K K DSβ φ

′= = = = −

′ ′ (8)

where , , andb bC Sφ β represents dimensionless substrate

concentration, the Thiele module, dimensionless parameter

for bulk fluid phase and substrate concentration in the bulk

fluid phase for reversible reaction. S represents the substrate

concentration inside the pellet, eqS is the equilibrium

substrate concentration, 'mV represents the maximum reaction

rate and 'mK is the Michaelis constants. The rate of change of

substrate concentration Y in batch reactor can be written as

01 b

dY Y

dt Yα

β=

+ (9)

where

00 0 0

0

, , ,b m bb b b eq

m mb

S a Ef V SY S S S

K KSα β

′−= = = = −

′ ′ (10)

where α represent the ratio of the catalyst volume to the

volume of the fluid phase reactor, ( 0.0252)a = is the ratio

of the catalyst volume to the volume of fluid phase in the

reactor. The Eqn. (9) is solved with the initial condition

( 0) 1Y t = = .

3. Analytical Expression of the

Concentration for Irreversible and

Reversible Reactions Using MADM

In the recent years, much attention is devoted to the

application of the modified Adomian decomposition method

(MADM) to the solution of various non-linear problems in

physical and chemical sciences. This method is used to find

the approximate analytical solution in terms of a rapidly



convergent infinite power series with easily computable

terms [17-18]. In other words, the zeroth component used in

the standard ADM can be divided into the two functions [19-

21]. The ADM is unique in its applicability, accuracy and

efficiency and only a few iterations are needed to find the

asymptotic solution. The basic concept of the method is

given in Appendix A.

3.1. Irreversible Reaction Without External Mass Transfer

Resistance

Solving equation (1) using this method (see Appendix B), we

obtain the concentration of the immobilized catalyst with g =1,

2, 3 for a slab, cylindrical, and spherical pellets as follows:

( ) ( )( )

( )2 4

2 4 2

3

1( ) 1 1 3 10 7

2 1 60 1b b

C X X X Xg

φ φβ β

= + − + − +

+ + (11)

Effective factor Ef for slab, cylindrical and spherical is

( )2

21

15 1 b

Efφ

β= −

+ (12)

Using Eqns. (7) and (12) the initial substrate reaction rate

0v can be obtained as follows:

( )2

00 2

0

115 1

m b

m bb

V Sv

K S

φβ

= − ++

(13)

3.2. Irreversible Reaction with External Mass Transfer

Resistance

The substrate concentration with external mass transfer

resistance for the initial and boundary conditions (Eqns. (2)

and (4)) is obtained (see Appendix C) from Eqn. (1) as

follows:

( )2 4

2 4 2

3

1 2 1 1 1 1 2 1( ) 1 1 ( 1) 1

2 (1 ) 20 3 2 51b b

C X X X Xg Bi Bi Bi Bi

φ φβ β

= + − − + − − + − − − + +

(14)

The change in reaction rate can be expressed quantitatively

by introducing the effectiveness factor, Ef . Using Eqn. (6),

effective factor for slab, cylindrical and spherical is

( )2

2

21

5 1 b

Efφβ

= −+

(15)

Using Eqns. (7) and (15), the initial substrate reaction rate

0v can be obtained as follows:

( )2

00 2

0

21

5 1

m b

m bb

V Sv

K S

φβ

= − ++

(16)

Summary of all the expression of substrate concentration

and effectiveness factor for with and without external mass

transfer resistance are also given in Table 1.

Table 1. Substrate concentration and effective factor in the with and without external mass transfer resistance.

Resistance Substrate concentration Fig. Effectiveness

factors Fig.

Initial

Reaction rate

Without external

mass

transfer resistance

( ) ( )( )

( )

2 42

3

4 2

11 1 60 1( ) 1

23 10 7

b b

X

C Xg

X X

φ φβ β

− +

+ += + − +

Eq. (11)

Fig. 1-3

8, 10(a) ( )2

21

15 1 b

Efφ

β= −

+

Eq. (12)

Fig.

9(a) ( )2

00 2

0

115 1

m b

m bb

V Sv

K S

φ

β

= − + +

Eq. (13)

With

external

mass

transfer

resistance

( )2 4

2

3

4

2

21

(1 ) 1

1 1 1 1 1( ) 1 ( 1)2 20 3 2

2 11

5

b b

XBi

C X Xg Bi

XBi Bi

φ φβ β

− − + + + = + − − + − − −

Eq. (14)

Fig. 4-6

9, 10(b) ( )2

2

21

5 1 b

Efφ

β= −

+

Eq. (15)

Fig.

9(b) ( )2

00 2

0

21

5 1

m b

m bb

V Sv

K S

φ

β

= − + +

Eq. (16)

3.3. Reversible Reaction

By replacing the variables , and by , ,b bC Cφ β φ β in

the Eqns. (11 -14), we can obtain the concentration of

substrate and effective factor of reversible reaction.

4. Analytical Expression of the

Concentration of Substrate Using the

New Approach of HPM

The advantage of the new homotopy perturbation method

(HPM) is that it does not need a small parameter in the


system [32]. Recently, many authors have used HPM for

various problems and reported the efficiency of the HPM to

handling nonlinear engineering problems [33-35]. Recently, a

new approach to HPM is introduced to solve the nonlinear

problem, in which one will get better simple approximate

solution in the zeroth iteration [19]. In this paper, a new

approach to the Homotopy perturbation method is applied

(Appendix D) to solve the nonlinear differential equation (9).

Using this method, the analytical expressions of the substrate

concentrations can be obtained as follows:

'

00 0

( ) exp exp1

b m

bb b

S a Ef VY t t t

S S

αβ

−= = − = +

(17)

5. Numerical Simulation

The non-linear differential equations (1) and (7) for the

given initial boundary conditions are solved numerically

using the Matlab program [36]. The numerical values of

parameters used in this work are given in Table 2 and Table

3. Its numerical solution is compared with our analytical

results in Tables (3) – (5) and it gives satisfactory agreement.

In all the case, the average relative error is less than 1.3%.

The Matlab program is also given in Appendix-E and F.

Table 2. Summarized description of the immobilized enzyme systems investigated [24].

Case no. 1 2 3 4

Enzyme Amyloglucocidase Amyloglucocidase Gulcoseisomerase Sweetzyme Q

Support Honey ceramic slab Porous spherical

Glass beads

Porous spherical

Glass beads

spherical

beads

reaction irreversible Irreversible reversible reversible

substrate Soluble starch Dextrin Glucose Glucose

Product Glucose Glucose Fructose Fructose

Reactor Stirred batch reactor Recycling differential batch

reactor Recycling differential batch reactor Stirred batch reactor

2( / )eD cm s 3.67 x 10-8 5.30 x 10-7 1.36 x 10-6 8.33 x 10-7

1 ( / )k cm s negligible Negligible 9.52 x 10-3 negligible

R (m) 1.6 x 10-4 1.6 x 10-4 1.6 x 10-4 3.2 x 10-5

mK 3( / )kg m 0.258 0.25042 - -

mV 3( / / )kg s m cat 2.51 x 10-2 0.4429 - -

( )mfK M - - 0.211 0.452

( / min/ )m fV mol lcat - - 0.1453 x10-3 0.142

( )mrK M - - 0.389 -

( / min/ )mrV mol lcat - - 2.783 x10-3 -

Figure 10 See in the subblimendary

material See in the subblimendary material

See in the

subblimendary material

References [39] [18] [40] [8]

Table 3. Numerical values of the parameter using this work.

Parameter Case 1 Case 2 Case 3 Case 4

Thiele Module / ( )m m eR V K Dφ =

Forward Backward Forward

/ ( )m m eR V K Dφ =



0.2605 2.1092 0.0035 0.0035 0.0197

Dimensionless parameter

in reaction for bulk fluid

phase

/b b mS Kβ = '/b b mfS Kβ = '/b b mrS Kβ = '/b b mfS Kβ =

337.054 56.65 412.132 223.54 192.38

Effectiveness factor

( )2

21

15 1 b

Efφ

β= −

+ 0.99 0.99 1 0.99 0.99



Table 4. Comparison of dimensionless substrate concentration C (without mass transfer resistance) with numerical result for various values of and bφ β .

(a) Spherical particle

g X 5φ = 10φ =

bβ Our Work Eqn. (11) Numerical % Error bβ Our Work Eqn. (11) Numerical % Error

3

0.0 2 0.161 0.163 1.22 15 0.105 0.107 1.86

0.2 5 0.406 0.413 1.69 20 0.277 0.282 1.77

0.4 7 0.581 0.581 0 25 0.470 0.470 0

0.6 10 0.762 0.762 0 50 0.791 0.791 0

0.8 15 0.907 0.907 0 100 0.940 0.940 0

1.0 10 1.000 1.000 0 500 1.000 1.000 0

Average deviation 0.582 Average deviation 0.728

g X 1bβ = 5bβ =

φ Our Work Eqn. (11) Numerical % Error φ Our Work Eqn. (11) Numerical % Error

3

0.0 4 0.282 0.288 2.08 6 0.211 0.215 1.86

0.2 3 0.463 0.465 0.43 5 0.413 0.416 0.72

0.4 2.5 0.636 0.636 0 4 0.644 0.644 0

0.6 2 0.807 0.807 0 3 0.843 0.843 0

0.8 1.5 0.935 0.935 0 2 0.960 0.960 0

1.0 0.1 1.000 1.000 0 1 1.000 1.000 0


(b) cylinder particle

g X 2φ = 10φ =


2

0.0 0.1 0.440 0.455 2.220 25 0.055 0.130 2.31

0.2 1 0.575 0.598 3.380 35 0.349 0.353 1.69

0.4 2 0.733 0.744 1.350 50 0.589 0.592 0

0.6 5 0.894 0.895 0 75 0.790 0.790 0

0.8 10 0.967 0.967 0 100 0.910 0.911 0

1.0 50 1.000 1.000 0 300 1.000 1.000 0


g X 1bβ = 5bβ =


2

0.0 3 0.300 0.306 1.96 5 0.202 0.206 1.94

0.2 2.5 0.454 0.458 0.87 4 0.432 0.438 1.36

0.4 2 0.655 0.655 0 3.5 0.587 0.587 0

0.6 1.5 0.829 0.829 0 3 0.765 0.765 0

0.8 1 0.955 0.956 0 2 0.940 0.940 0

1.0 0.1 1.000 1.000 0 1 1.000 1.000 0


(c) Slab particle

g X 1.5φ = 5φ =


1

0.0 1 0.554 0.564 1.773 20 0.428 0.431 0.696

0.2 2 0.670 0.681 1.615 25 0.540 0.549 1.630

0.4 3 0.767 0.770 0.518 30 0.662 0.662 0

0.6 5 0.880 0.883 0 50 0.843 0.843 0

0.8 10 0.963 0.963 0 100 0.955 0.955 0

1.0 15 1.000 1.000 0 500 1.000 1.000 0



g X 1bβ = 5bβ =


1

0.0 1.5 0.554 0.564 1.77 3 0.271 0.373 2.70

0.2 1.4 0.596 0.607 1.90 2.5 0.509 0.519 1.96

0.4 1.2 0.705 0.709 0.50 2 0.723 0.735 1.36

0.6 1 0.843 0.853 1.76 1.5 0.880 0.880 0

0.8 0.7 0.956 0.956 0 1.2 0.956 0.956 0

1.0 0.01 1.000 1.000 0 0.1 1.000 1.000 0


Table 5. Comparison of dimensionless substrate concentration C (with mass transfer resistance) with numerical results for various values of , , andb iBφ β .

(a) Slab Particle

g X 1, 5Biφ = = 1, 1bφ β= =

bβ

Our Work Eqn. (13) Numerical % Error Bi Our Work Eqn. (13) Numerical % Error

1

0.0 1 0.764 0.765 0.13 5 0.752 0.761 1.31

0.2 1.5 0.754 0.755 1.43 5.5 0.809 0.813 1.23

0.4 2.5 0.835 0.835 0 6 0.845 0.845 0

0.6 5 0.913 0.913 0 7 0.897 0.897 0

0.8 10 0.965 0.965 0 8 0.957 0.957 0

1.0 15 0.987 0.987 0 15 0.993 0.993 0


(b) Cylinder Particle

g X 1, 0.5Biφ = = 1, 2bφ β= =

bβ


2

0.0 2 0.637 0.6414 1.24 0.5 0.640 0.641 0.15

0.2 3 0.704 0.715 .071 0.75 0.725 0.727 0.27

0.4 4 0.765 0.765 0 1 0.784 0.784 0

0.6 6 0.836 0.836 0 2 0.867 0.867 0

0.8 10 0.901 0.901 0 5 0.937 0.937 0

1.0 15 0.937 0.937 0 10 0.983 0.983 0


(c) Spherical Particle

g X 1, 1Biφ = = 1, 1bφ β= =

bβ


3

0.0 0.1 0.661 0.664 0.45 0.5 0.650 0.658 1.21

0.2 0.5 0.721 0.726 0.69 1 0.775 0.778 0.38

0.4 1 0.787 0.787 0 1.5 0.832 0.832 0

0.6 2 0.859 0.859 0 2 0.871 0.871 0

0.8 10 0.964 0.964 0 5 0.938 0.938 0

1.0 100 0.996 0.996 0 20 0.991 0.991 0


6. Result and Discussion

Eqns. (11-12) and (14-15) represent the analytical

expression for the dimensionless substrate concentration

( )C X and effectiveness factor Ef for both reversible and

irreversible reactions without and with mass transfer

resistance for slab, cylinder and spherical pellets respectively.

The substrate concentration C against the dimensionless

radial distance X for the both reversible and irreversible

reactions is plotted in Figs. 2 – 7 for various values of the

Thiele modulus and bφ β for the three shapes. When the

Thiele modulus or Half – thickness of the pellet ( )R

increases, the substrate concentration inside catalyst will also

decrease in all the cases.

Figs. 2 (a) - 4 (a) represents that the substrate concentration

C versus distance X for various values of and bφ β without

external mass transfer resistance. The substrate concentration

increases with increasing bβ or increasing irreversible

substrate concentration in the bulk fluid phase ( )bS and

decreasing the irreversible reaction Michaelis constant ( )mK .

From these figures, it is obvious that the substrate

concentration reaches a uniform value when the Thiele

modulus 0.01φ ≤ and 200bβ > . Figs. 2 (b) – 4 (b) shows

that, the substrate concentration decreases with increasing the

Thiele modulus φ or increasing Half – thickness of the pellet

( )R and irreversible maximum reaction rate ( )mV .



(a). 1.5φ = and 1 to 50bβ = (b). 5bβ = and 0.1 to 3φ =

Figure 2. Plot the dimensionless substrate concentration C versus dimensionless distance X , in the slab pellet calculated using Eqn. (11).

(a) 10φ = and 25 to 300bβ = (b). 1bβ = and 0.1to3φ =

Figure 3. Plot the dimensionless substrate concentration C versus dimensionless distance X , in the cylinder pellet calculated using Eqn. (11).

(a). 5φ = and 2 to 100bβ = (b). 1bβ = and 0.1to 4φ =

Figure 4. Plot the dimensionless substrate concentration C versus dimensionless distance X , in the spherical pellet calculated using Eqn. (11).


From Figs. (5-7) it is inferred that substrate concentration increases with the increasing Biot number ( )iB and Michaelis-

Menten constant ( )bβ . The dimensionless substrate concentrations for the three pellets are plotted in Figs. (8). From these

figures, it is concluded that the dimensionless substrate concentration for the spherical pellets is greater than slab and

cylindrical pellets.

(a). 1 , 1bφ β= = and 1 to 15Bi = (b). 1, 5Biφ = = and 1 to 10bβ =

Figure 5. Plot the dimensionless substrate concentration C versus dimensionless distance X , in the slab pellet calculated using Eqn. (14).

(a). 1 , 2bφ β= = and 0.5 to 5Bi = (b). 1, 0.5Biφ = = and 2 to 10bβ =

Figure 6. Plot the dimensionless substrate concentration C versus dimensionless distance X , in the cylinder pellet calculated using Eqn. (14).

(a). 1 , 1bφ β= = and 0.5 to 20Bi = (b). 1, 1Biφ = = and 0.1 to100 .bβ =

Figure 7. Plot the dimensionless substrate concentration C versus dimensionless distance X , in the spherical pellet calculated using Eqn. (14).



Figure 8. Plot of dimensionless substrate concentration C versus the dimensionless distance X for the slab, cylindrical, spherical pellets (a) without mass-

transfer resistance (Eqns. (11)), (b) with mass transfer resistance (Eqn. (14)).

Plot of effectiveness factor Ef against Thiele modulus φ and Michaelis - Menten constant bβ is shown in Fig. 9 (a - b).

Effectiveness factor is a dimensionless pellet production rate that measures how effectively the catalyst is being used. For η

near unity, the entire volume of the pellet is reacting at the same high rate because the reactant is able to diffuse quickly

through the pellet. For η near zero, the pellet reacts at low rate. The reactant is unable to penetrate significantly into the interior

of the pellet and the reaction rate is small in a large portion of the pellet volume. The effectiveness factor decreases from its

initial value, when the diffusional restriction or bβ increases. The effectiveness factor is maximum ( 1)Ef ≈

at lower values of

and bφ β . For all the cases, 1Ef ≈ .

(i) (a-b) Without mass transfer resistance using ( Eqn. (12)), (ii) (a-b) With mass transfer resistance using (Eqn. (15)).

Figure 9. The general effectiveness factor Ef against Thiele modulus φ and Michaelis-Menten constant bβ .


Now the Eqn. (7) can be written as follows 0 0 0/ ( / ) ( / )b m m b mS v K V S V= + . The plot of Fig. 10 represents 0 0/bS v versus

irreversible initial substrate concentration in the bulk fluid phase 0bS gives the slope = 1/ mV and the intercept= /m mK V . The

parameters andm mK V are obtained from the above slope and intercept results. Good agreement between predicted and

experimental data is observed at low initial substrate concentrations.

(a). Hanes-Woolf plot (b). Line weaver- Burk plot

Figure 10. Comparison of our analytical result (Eqn. (13)) initial substrate reaction rate with the numerical result [24] and experimental data [24] (Refer

Table 3) for case 1.

Our analytical expression (Eqn. (17)) for the concentration of substrate 0( ( ) )b bS Y t S= is compared with the experimental

results in Fig. (11). Good agreement with the experimental data is noted. From this figure, it is inferred that reversible substrate

concentration bS is almost uniform when the initial bulk substrate concentration 0bS is constant. The substrate concentration

in the fluid phase bS increases when the initial bulk substrate concentration 0bS increases.

Figure 11. Comparison of our analytical result (Eqn. (17)) for substrate concentration with the numerical result [24] and experimental data [24] for the time

courses of substrate consumption in a batch reactor model (Refer Table 3).

7. Conclusions

In this paper an approximate analytical solutions of the

nonlinear initial boundary value problem in Michaelis-

Menten kinetics have been derived. The modified Adomian

decomposition method (MADM) is used to obtain the

solutions for the non-linear model of an immobilized

biocatalyst enzyme. Approximate analytical expressions for

the concentration of substrate and the effective factor in

immobilized biocatalyst enzymes are derived. The analytical

solutions agree with the experimental results and numerical

solutions (Matlab program) with and without external mass

resistance for a slab, cylindrical and spherical pellets. These



analytical results are more descriptive and easy to visualize

and optimize the kinetic parameters of immobilized enzymes.

Appendix A: Basic Concept of Modified

Adomian Decomposition Method

Consider the singular boundary value problem of 1n +

order nonlinear differential equation in the form

( ) ( )

( ) ( )

1

10 1 1

( ),

(0) , (0) ,. . . , 0 ,

n n

nn

my y N y g x

x

y a y a y a y b c

+

−−

+ + =

′= = = = (A.1)

Where N is a non-linear differential operator of order less

than n , ( )g x is given, function and 0 1 1, ,... , ,na a a c b− are

given constants. We propose the new differential operator, as

below

( )1 1 .n

n m m n

n

d dL x x x

dxdx

− + − −= (A.2)

Where , 1m n n≤ ≥ , so, the problem can be written as

( ) ( )1 .L g x Ny− = − (A.3)

The inverse operator 1L−

is therefore considered a 1n +

fold integral operator, as below

( ) ( )1 1

0 0 0

. .... . ... .

x x x x

n m m n

b

L x x x dx dx− − − −= ∫ ∫ ∫ ∫ (A.4)

By applying 1L−

on (A.3), we have

( ) ( ) ( )1 1y x x L g x L Nyϕ − −= + (A.5)

Such that ( ) 0L xϕ =

The Adomian decomposition method introduces the

solution ( )y x and the nonlinear function Ny by infinite

series

( ) ( )0

n

n

y x y x

∞

=

=∑ (A.6)

and

0

n

n

Ny A

∞

=

=∑ (A.7)

where the components ( )ny x of the solution ( )y x will be

determined recurrently. Specific algorithms were seen in [8,

12] to formulate Adomian polynomials. The following

algorithm:

( )0 ,A F u=

( )1 0 1,A F u u=

( ) ( )'' 22 0 2 0 1

1,

2A F u u F u u= +

( ) ( )'' 22 0 2 0 1

1,

2A F u u F u u= +

( ) ( )'' 22 0 2 0 1

1,

2A F u u F u u= +

( ) ( ) ( )'' ''' 33 0 3 0 1 2 0 1

1 1, ,

2 3!A F u u F u u u F u u= + + (A.8)

can be used constant Adomian polynomials, when ( )F u

is a nonlinear function. By substituting (A. 6) and (A. 7) into

(A. 5)

( ) ( )1 1

0 0

n n

n n

y x L g x L Aϕ∞ ∞

− −

= =

= + −∑ ∑ (A.9)

Through using the modified Adomian decomposition

method, the components ( )ny x can be determined as

10

11

( ) ( )

( ) ( ), 0n n

y x A L g x

y x L A n

−

−+

= +

= − ≥ (A.10)

which gives

10

11 0

12 1

13 2

( ) ( )

( ) ( )

( ) ( )

( ) ( )

...

y x A L g x

y x L A

y x L A

y x L A

−

−

−

−

= +

= −

= −

= −

(A.11)

From (A.8) and (A.11), we can determine the components

( )ny x , and hence the series solution of ( )y x in (A.6) can be

immediately obtained. For numerical purposes, the n- term

approximate

1

0

n

n k

n

yψ−

=

=∑ (A.12)

can be used to approximate, the exact solution. The approach

presented above can be validated by testing it on a variety of

several linear and nonlinear initial value problems.

Appendix B: Analytical Solution of

Substrate Concentration Without

External Mass Transfer Resistance

The solutions of Eq. (1) for 3g = allow us to predict the

concentration profiles of dimensionless substrate

concentration in immobilized enzymes. In order to solve Eq.


(1), using the modified Adomian decomposition method, Eq.

(1) can be written with the operator form

20

01 b

CLs

C

φβ

= +

(B.1)

where2

2

dL

dx= , Applying the inverse operator

1L− on both

sides of Eq. (B. 1) yields

( )2

0

01 b

CC x Ax B

C

φβ

= + + +

(B.2)

Where A and B are the constants of integration. We let,

( )0

n

n

C x C

∞

=

=∑ (B.3)

( )0

n

n

N C x A

∞

=

= ∑ (B.4)

Where

( )2

0

01 b

CN C x

C

φβ

= +

(B.5)

From the eqns (B. 3), (B. 4) and (B. 5), Eq. (B. 2) gives

( )2

0

001

nbn

CC x Ax B

C

φβ

∞

=

= + + +

∑ (B.6)

We identify the zeroth component as

0 ( )C x AX B= + (B.7)

And the remaining components as the recurrence relation

2 11 ; 0n nC L A nφ −

+ = ≥ (B.8)

where nA are the Adomian polynomials of 0, 1, ... , nC C C .

We can find the first few nA as follows:

Apply the boundary conditions in (B. 1) we get,

0 1C = (B.9)

Again to find 1C

21 0

101 b

CC L

C

φβ

− = +

(B.10)

Using (B. 9) in (B. 10),

21

11 b

C Lφ

β−

= + (B.11)

Again using this formula to find 1C ,

21

1

0 01

X X

b

C X X dXdXφ

β−

= + ∫ ∫ (B.12)

Integrating Eqn. (B. 12),

2 21

11 6b

XC A BX

φβ

− = + + +

(B.13)

Where A and B are integrating constants. Again using

boundary conditions Eqn. (B. 13) becomes,

( ) ( )2

21 1

6 1 b

C Xφ

β= −

+ (B.14)

Now, consider

( )12 0 1

0

1

1!

dC L N C C

d λλ

λ−

=

= +

(B.15)

Solving ( )0 1

0

1

1!

dN C C

d λλ

λ =

+

we get,

( )( )

41 2

2 31

6 1 b

C L Xφ

β− = − +

(B.16)

Therefore,

( )( )

41 2

2 31

6 1 b

C X X X dXdXφ

β−

= − +

∫∫ (B.17)

Integrating Eqn. (B. 17),

( )4 4 3

12 3 120 361 b

X XC C DX

φβ

− = − + + +

(B.18)

Where C and D are integrating constants. Apply boundary

conditions we get the value for C and D,

Therefore Eqn. (B. 17) in the form,

( )( )

44 2

2 33 10 7

360 1 b

C X Xφ

β= − +

+ (B.19)

Adding the Eqns. (B. 9), (B. 14) and (B. 19) we get the

solution Eqn. (7). Similarly, to apply the above method for

1, 2g g= = to find the solution.

Appendix C: Analytical Solution of

Substrate Concentration with External

Mass Transfer Resistance

The solutions of Eq. (1) for 3g = allow us to predict the

concentration profiles of dimensionless substrate

concentration in immobilized enzymes. In order to solve Eq.



(1), using the modified Adomian decomposition method, Eq.

(1) can be written with the operator form

20

01 b

CLs

C

φβ

= +

(C.1)

Where 2

2

dL

dx= , Applying the inverse operator

1L− on

both sides of Eq. (C.1) yields

( )2

0

01 b

CC x Ax B

C

φβ

= + + +

(C.2)

Where A and B are the constants of integration. We let,

( )0

n

n

C x C

∞

=

=∑ (C.3)

( )0

n

n

N C x A

∞

=

= ∑ (C.4)

Where

( )2

0

01 b

CN C x

C

φβ

= +

(C.5)

From the eqns (C.3), (C.4) and (C.5), Eq. (C.2) gives

( )2

0

001

nbn

CC x Ax B

C

φβ

∞

=

= + + +

∑ (C.6)

We identify the zeroth component as

0 ( )C x AX B= + (C.7)

And the remaining components as the recurrence relation

2 11 ; 0n nC L A nφ −

+ = ≥ (C.8)

where nA are the Adomian polynomials of 0, 1, ... , nC C C .

We can find the first few nA as follows:

C AX B= + (C.9)

Apply the boundary conditions in (C. 9) we get,

0 1C = (C.10)

Again to find 1C

21 0

101 b

CC L

C

φβ

− = +

(C.11)

Using (C. 10) in (C. 11),

21

11 b

C Lφ

β−

= + (C.12)

Again using this formula to find 1C ,

21

1

0 01

X X

b

C X X dXdXφ

β−

= + ∫ ∫ (C.13)

Integrating Eqn. (C. 13),

2 21

11 6b

XC A BX

φβ

− = + + +

(C.14)

Where A and B are integrating constants. Again, using

boundary conditions Eqn. (C. 14) becomes,

2 2

1

1 1

1 6 3 6b

XC

Bi

φβ

= − − +

(C. 15)

Now, consider

( )12 0 1

0

1

1!

dC L N C C

d λλ

λ−

=

= +

(C.16)

Solving ( )0 1

0

1

1!

dN C C

d λλ

λ =

+

we get,

( )4 2

12 3

1 1

6 3 61 b

XC L

Bi

φβ

− = − − +

(C.17)

Therefore,

( )4 2

12 3

1 1

6 3 61 b

XC X X dXdX

Bi

φβ

− = − − + ∫∫

(C.18)

Integrating Eqn. (C. 18),

( )4 4 2

12 3

1 1

120 6 3 61 b

X XC C DX

Bi

φβ

− = − + + + +

(C.19)

Where C and D are integrating constants. Apply boundary

conditions we get the value for C and D,

Therefore Eqn. (C. 11) in the form,

( )4 2

42 3

1 1 1 1 1 1( 1)

120 3 6 6 6 3 301 b

XC X

Bi Bi Bi

φβ

= − − + − − − +

(C.20)

Adding the Eqns. (C. 10), (C. 15) and (C. 20) we get the

solution Eqn. (13). Similarly, to apply the above method for

1, 2g g= = to find the solutions.


Appendix D1: Scilab Program for the

Numerical Solution of Equation (11)

function pdex4

m = 0;

x=linspace(0, 1);

t = linspace(0, 10000000);

sol = pdepe(m, @pdex4pde, @pdex4ic, @pdex4bc, x, t);

u1 = sol(:, :, 1);

%————————————————————–

figure

plot(x, u1(end, :))

title('u1(x, t)')

xlabel('Distance x')

ylabel('u1(x, 1)')

function [c, f, s] = pdex4pde(x, t, u, DuDx);

c=1;

f =1.* DuDx;

a=6; b=5;

F =-((a^2*u(1))/((1+b*u(1))));

s =F;

% ————————————————————–

function u0 = pdex4ic(x);

u0 = [0];

% ————————————————————–

function [pl, ql, pr, qr] = pdex4bc(xl, ul, xr, ur, t)

pl = [0];

ql = [1];

pr = [ur(1)-1];

qr = [0];

Appendix D2: Scilab Program for the

Numerical Solution of Equation (13)

function pdex4

m = 0;

x = linspace(0, 1);

t = linspace(0, 1000000);

sol = pdepe(m, @pdex4pde, @pdex4ic, @pdex4bc, x, t);

u1 = sol(:, :, 1);

% ————————————————————–

figure

plot(x, u1(end,:))

title('u1(x, t)')

xlabel('Distance x')

ylabel('u1(x, 1)')

function [c, f, s] = pdex4pde(x, t, u, DuDx)

c =1;

f =1.* DuDx;

e=1; alpha=5;

F =-(e*u(1))/((1+(alpha*u(1))));

s =F;

% ————————————————————–

function u0 = pdex4ic(x)

u0 = [0];

% ————————————————————–

function [pl, ql, pr, qr] = pdex4bc(xl, ul, xr, ur, t)

B=1;

pl = [0];

ql = [1];

pr = [-B*(1-ur(1))];

qr = [1];

(i). Michaelis-Menten constant bβ , (ii). Thiele modulus φ .

Figure 12. Plot of the three-dimensional dimensionless concentration C against the dimensionless distance X for the three pellets calculated using Eqn. (11)

(without mass-transfer resistance).



(i). Michaelis-Menten constant bβ , (ii). Thiele modulus φ

Figure 13. Plot of the three-dimensional dimensionless concentration C against the dimensionless distance X for the three pellets calculated using Eqn. (14)

(with mass-transfer resistance).

Figure 14. Hanes-Woolf plot for case 2 in the absence of mass transfer limitations (dotted line), model predictions with constant diffusivity (dot-dashed line),

model predictions with concentration-dependent diffusivity (dashed line), experimental data (symbols), and analytical result (solid line).

Figure 15. Line weaver-Burk plot for case 2; model predictions using the optimized parameters (dotted line), experimental data (symbols) and analytical

result (solid line).


Figure 16. Line weaver-Burk plot for the forward reaction of case 3; model predictions using the optimized parameters (dotted line), experimental data

(symbols) and analytical result (solid line).

Figure 17. Line weaver-Burk plot for the forward reaction of case 4; model predictions using the optimized parameters (dotted line), experimental data

(symbols) and analytical result (solid line).

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